Does the Ross Recovery Theorem work Empirically?

Transcription

1 Does the Ross Recovery Theorem work Empirically? Jens Carsten Jackwerth Marco Menner June 24, 206 Abstract Starting with the fundamental relationship that state prices are the product of physical probabilities and the pricing kernel, Ross (205) shows that, given severe assumptions, knowing state prices suffices for backing out physical probabilities and the pricing kernel at the same time. We empirically investigate the performance of such recovery based on the S&P 500 index by testing if future, realized returns are indeed compatible with the recovered physical distributions. Statistical tests are strongly rejected in our sample and we document a number of empirical problems. Keywords: Ross recovery, Berkowitz, pricing kernel, risk-neutral density, physical probabilities Jens Jackwerth is from the University of Konstanz, PO Box 34, Konstanz, Germany, Tel.: +49-(0) , Fax: +49-(0) , jens.jackwerth@uni-konstanz.de Marco Menner is from the University of Konstanz, PO Box 34, Konstanz, Germany, Tel.: +49- (0) , Fax: +49-(0) , marco.menner@uni-konstanz.de We received helpful comments and suggestions from Guenther Franke. We thank seminar participants at the University of Konstanz, the University of Strasbourg, the University of Zurich, the University of St. Gallen, the University of Sydney, and the University of Queensland.

2 I. Introduction Much of financial economics revolves around the triangular relation between physical return probabilities p, which are state prices π divided by the pricing kernel m: physical probability p = state price π pricing kernel m () Present research picks any two variables to find the third. In option pricing, for example, physical probabilities are changed into risk-neutral ones, which are nothing but normalized state prices, via the pricing kernel. Differently, the pricing kernel puzzle literature, e.g. as Cuesdeanu and Jackwerth (206), starts out with risk-neutral and physical probabilities in order to find the empirical pricing kernel. Yet Ross (205) presents a recovery theorem which allows to back out both the pricing kernel and physical probabilities by only using state prices. To achieve this amazing feat, he needs to make strong assumptions concerning the economy. Our contribution is to empirically analyze his theorem and test if the recovered physical probabilities are compatible with future, realized S&P 500 returns. We further analyze if the shape of the recovered pricing kernel is in line with utility theory. To understand our sobering results, we detail the underlying assumptions of Ross (205) some more. First, the Ross recovery theorem requires time-homogeneous transition state prices π i,j that represent state prices of moving from any given state i today to any other state j in the future. Such transition state prices include the usual spot state prices π 0,j with 0 representing the current state of the economy. Spot state prices can readily be found from option prices, see e.g. Jackwerth (2004). Yet Ross recovery also requires the transition state prices emanating from alternative, hypothetical states of the world. 2 There is, however, no market with actively traded financial See e.g. Cochrane (2000), pp. 50. A state (Arrow-Debreu) price represents the amount of dollars an investor is willing to pay for a security that pays out one dollar if the particular state occurs and nothing if any other state occurs. The pricing kernel is closely related to the marginal utility of a representative agent. 2 Imagine that the current state of the world is characterized by the S&P 500 being at 000. Let there be two future states, 900 and 000, to which the spot state prices (emanating from 000) relate. The required other transition state prices are the ones emanating (hypothetically) from 900 and ending at 900 or 000 2

3 products which allow to back out (non-spot) transition state prices π i,j directly. We therefore use information on spot state prices with N different maturities to obtain transition state prices linking N states. 3 This requires us to first estimate a rich spot state price surface from sparse option data. There is no unique way for estimating transition state prices π i,j from spot state prices, and so we suggest several different methods. We also introduce a version that allows to apply the recovery theorem directly to spot state prices π 0,j. Second, all transition state prices need to be positive, which turns out to be a fairly benign assumption as we can always force some small positive state price. 4 Finally, the pricing kernel is restricted to be a constant times the ratio of values in state j over values in state i. A convenient economic interpretation of these values is to associate them with marginal utilities in those states. The constant can then be associated with a utility discount factor. Taken together, the three assumptions allow Ross (205) to formulate a unique eigenvalue problem. Its solution yields the physical transition probabilities p i,j, which represent physical probabilities of moving from state i to state j, and the pricing kernel. After we achieve recovery, we empirically test the hypothesis that future, realized S&P 500 returns are drawn from the recovered physical spot distribution p 0,j. In each month t, we work out the percentile of next month s S&P 500 return based on the recovered physical spot cumulative distribution function. This leaves us with one value x t for each month t that lies in between 0 and. If our hypothesis holds, the set {x t } is uniformly distributed. Following Berkowitz (200) 5, we can strongly reject our hypothesis. Ross (205) does not recover physical spot distributions of returns, which are consistent with future, realized S&P 500 returns. We further find that the recovery theorem does not produce reasonable pricing kernels as they are riddled with local minima and maxima. In contrast, we cannot reject the hypotheses that future 500 returns are drawn from one period later. 3 For implementation details and variations, see below. 4 Technically, some zero values could be allowed as long as any state can still be reached from any other state, possibly via some other intermediate states, Ross (205) 5 Note that Berkowitz (200) does not directly test for uniformity of the values but for standard normality of a standard normal transformation of the values. 3

4 simple physical distributions based on a power pricing kernel or the five-year historical return distribution. The empirical problems of Ross recovery stem from three sources. For one, the numerical methods for obtaining transition state prices from (noisy) option prices add estimation error to the state prices. Second, the assumption of time-homogeneous transition state prices might not hold and different periods might require different transition state prices. Third, the strong assumption concerning the functional form of the pricing kernel is rather limiting. Note that standard CARA or CRRA pricing kernels are incompatible with Ross recovery as his multi-period pricing kernels are identical but for a constant, while, under CARA or CRRA, they are multiplications of single-period pricing kernels. 6 Only a few papers already investigate the recovery theorem from an empirical perspective. Closest to our work are Audrino et al. (205), who also implement Ross recovery on S&P 500 index options while staying close to the original methods of Ross (205). They differ from Ross (205) and us in that they use a neural network to interpolate the state price surface instead of smoothing techniques. Surprisingly, their recovered pricing kernels tend to be rather smooth and u-shaped, as opposed to our wavy pricing kernels. The subtle reason is a quadratic penalty term which they use to force all transition state prices to zero, but which, empirically, is stronger for states further away from the current states as option prices are more sensitive to state prices around the current state. As a result, the implied risk-free rates (= - sum of transition state prices) increase in distance to the current state, which in turn mechanically leads to u-shaped pricing kernels. Their further empirical focus is on developing profitable trading strategies based on the recovered physical probabilities, but, unlike our work, they do not statistically test if future, realized returns are drawn from the recovered distribution. As an alternative to the numerically difficult recovery of transition state prices from spot state prices, we suggest an additional, implicit method, which obviates that recovery and works directly 6 Moreover, under Ross recovery, pricing kernels are scaled differently according to each initial state, while CRRA pricing kernels (using returns as states) and CARA pricing kernels (using prices as states) do not depend on the initial state. 4

5 with the spot state prices. Jensen et al. (206) suggest the same method and further restrictions of the pricing kernel. Their focus is then on using the mean of the recovered physical probabilities to predict S&P 500 return with a small yet significant R 2 of.28%. While they also apply a Berkowitz test and reject their particular model, they, in contrast to our work, do not cover the original formulation of Ross (205). Informing our empirical work is a simulation study by Tran and Xia (204), who show that the recovered physical probabilities vary substantially for different state space dimensions. We check for this prediction in our robustness tests but it empirically does not matter much for our paper. Related to Ross recovery, Hansen and Scheinkman (2009) already showed theoretically that the Perron-Frobenius theorem can be used to identify a probability measure p from a risk-neutral process, where p contains information on long-term risk. Yet they argue that this measure typically differs from the true physical physical measure p. Borovicka et al. (205) build on that approach and show that Ross, who recovers that probability measure p, ignores a non-trivial martingale component that links the pricing kernel associated with p and the pricing kernel associated with the true physical probability measure p. Bakshi et al. (205) support this result with the empirical finding that this martingale component indeed exists. Yet, these papers point out that the recovered probability measure might be useful in analyzing long-term bond prices. Turning this argument around, Martin and Ross (203) show that the recovered pricing kernel can be inferred from the time series behavior of the long bond and information on the long end of the yield curve. While Ross recovery works on a discrete state space using the Perron-Frobenius theorem, Carr and Yu (202) show that recovery can be achieved in continuous time for univariate timehomogeneous bounded diffusion process by using Sturm-Liouville theory. Walden (204) further investigates an extension of the recovery theorem to continuous time if the diffusion process is unbounded. He derives necessary and sufficient conditions that enable recovery and finds that recovery is still possible for many of these unbounded processes. Qin and Linetsky (205) extend the recovery theorem to continuous-time with state variables that follow a Borel-right process. This class contains relevant models in finance such as multi-dimensional diffusion models and affine jump 5

6 models for both bounded and unbounded state spaces. Dubynskiy and Goldstein (203) explain that bounding the state variable process is the key mechanism that makes the recovery of drifts and risk aversion possible. They argue that such boundary conditions have a significant impact on the recovery result and should only be applied if they can be justified economically. Further work on the recovery of subjective probabilities and the pricing kernel is done by Schneider and Trojani (205). They express the pricing kernel as a polynomial function of both riskneutral and subjective moments. They minimize its expected variance and require mild economical assumptions such as a negative divergence premium and bounded conditional subjective moments to achieve recovery. The remainder of the paper proceeds as follows. Section 2 explains the Ross recovery theorem. In Section 3, we introduce our methods to obtain spot state prices, to back out transition state prices, and to apply the theorem without using transition state prices. We further explain the Berkowitz test, which we use as the main test in our empirical approach, as well as additional statistical tests which we apply for robustness. Section 4 describes our data set. In section 5, we present the empirical results of our study. Section 6 provides several robustness checks. Section 7 contains concluding remarks. II. The Ross Recovery Theorem An application of the recovery theorem requires the following assumptions: (i) Transition state prices π i,j need to be strictly positive, (ii) the transition state prices follow a time homogeneous process, which means that they are independent of calender time, and (iii) the corresponding pricing kernel m i,j is transition independent, which means that it can be written as: m i,j = δ u j u i (2) for a positive constant δ and positive state dependent (marginal utility) values u j and u i. With this structure for the pricing kernel, the physical transition probabilities p i,j have the 6

7 form: p i,j = π i,j m i,j = δ πi,j u i u j (3) The Ross recovery theorem then allows to uniquely determine δ and all the u i from the transition state prices π i,j and thus can uniquely solve for the physical transition probabilities. We illustrate the recovery theorem in a simple example with two states, state 0 and state. For any of the two possible initial states, the physical transition probabilities have to sum up to one: p 0,0 + p 0, = δ π 0,0 u 0 u 0 p,0 + p, = δ π,0 u u 0 + δ π 0, u 0 u + δ π, u u We can rewrite this system of equations in matrix form and obtain the following eigenvalue problem: = = (4) π 0,0 π 0, z 0 = δ z 0 where z 0 = u π,0 π 0, z z and z = u (5) Given assumptions (i), (ii), and (iii), an application of the Perron-Frobenius theorem leads to the result that there is only one eigenvector z with strictly positive entries z 0 and z. That eigenvector corresponds to the largest (and positive) eigenvalue δ of the eigenvalue problem. This property implies a unique positive pricing kernel m i,j as in equation (2) and unique physical transition probabilities p i,j for i, j = 0,. In the general setting with N different states, Ross defines the state transition matrix Π with entries π i,j. Each row i in Π represents state prices of moving from a particular state i to any other state j. We always label the current state with i = 0 out of a set I = { N low,..., 0,..., N high } where N = N low + N high +. States lower than the current state are indexed with a negative i, while states higher than the current state are indexed with a positive i. The ending transition 7

8 state j is drawn from the same set I. Analogous to the two state example, we solve the following N-dimensional eigenvalue problem: Πz = δz, where z i = u i (6) Knowing z i and δ, we are then able to recover the physical transition probabilities p i,j. The 0 th row of Π represents the spot state prices π 0,j, which coincide for the first period with the transition state prices. III. Methodology We now explain how we estimate a smooth implied volatility surface based on observed option prices. We then transform this smooth implied volatility surface into a smooth call option price surface. From this call option price surface, we obtain spot state prices π (t) 0,j for different maturities t, which we then use to back out transition state prices π i,j. A. Smoothing the Implied Volatility Surface We use European put- and call options on the S&P 500 for our analysis and transform observed bidand ask option prices into implied volatility bid- and ask quotes. Quotes are only available for some specific states and maturities, but we are interested in a rich and smooth implied volatility surface. Jackwerth (2004) introduces the fast and stable method to find smooth implied volatilities σ i on a fine grid of states i for a fixed maturity. He minimizes the sum of squared second derivatives of implied volatilities (insuring smoothness of the volatility smile) plus the sum of squared differences between model and observed implied volatilities (insuring the fit to the options data) using a trade-off parameter λ. We extend the method to volatility surfaces by adding a maturity dimension to the S&P 500 level dimension. Again, we minimize the sum of squared local total second implied volatility derivatives σ i,t (insuring smoothness of the volatility surface and not only the volatility smile) plus the sum of 8

9 squared violations of the observed bid and ask implied volatilities (insuring the fit to the options data by penalizing model values outside the bid/ask bounds) using a trade-off parameter λ. The optimization problem is then defined as: min σ i,t, δ bid l, δl ask T N T N t= i= (σ i,t) 2 + λ L s.t. L l= ( δ bid l ) 2 + δl ask δ ask l σ il,t l σ ask i l,t l δ bid l σ bid m l,t l σ il,t l σ i,t, δl bid, δl ask, 0 (7) where σ bid i l,t l and σ ask i l,t l observations. δ bid l δ ask l are the l th observed implied volatility bounds with a total number of L represents the distance of the implied volatility, if it is below the bid bound and represents the distance of the implied volatility, if it is above the ask bound. One advantage of that method is that implied volatility can move flexible within bid- and ask bounds. We transform the resulting implied volatilities into call option prices and, by means of Breeden and Litzenberger (978) and for each maturity t, take the numerical second derivative of call prices to obtain spot state prices. We insure positivity and smoothness of the spot state price surface by reducing the trade-off parameter λ in (7) until our imposed conditions for smoothness and positivity of the state price surface are fulfilled. Appendix A provides details for these conditions and the implementation. We discretize the maturity at a step size of 3.5 days and let it move within a range of 6.5 days and 365 days. We round observed option expiry dates to the respective next maturity that lies on that grid. For the state space dimension, we discretize option strike prices with a step size of $5. We then convert strike prices into moneyness by normalizing them with the current level of the S&P 500 index. This discretization enables to use all available information on option prices within the defined maturity range. We set sufficient upper and lower bounds for the moneyness such that all possible states with a non-zero probability of occurrence are covered. Figure shows the smoothed implied volatility surface on February 7, 200, while Figure 2 illustrates spot state prices based on the smoothed implied volatility surface. 9

10 Volatility Surface, 7-Feb Implied Volatility Moneyness Maturity in days Figure. Implied volatility surface obtained by our smoothing method on February 7, 200. Bid- and ask observed implied volatilities are marked as black dots and gray squares. State Price Surface, 7-Feb Spot State Price Surface Moneyness Maturity in days Figure 2. Spot state prices implied by the smoothed implied volatility surface in Figure on February 7,

11 One-month implied volatilities (Panel A) and the corresponding one-month state prices (Panel B) are shown in Figure 3. Observed bid- and ask implied volatilities are marked with black dots and gray squares. 0.5 Panel A: 7-Feb Implied Volatility Moneyness Panel B: 7-Feb-200 Spot State Prices Moneyness Figure 3. Implied volatilities obtained by our smoothing method (Panel A) and the related spot state prices (Panel B) for a given maturity of 30 days on February 7, 200. Bid- and ask observed implied volatilities are marked as black dots and gray squares. B. Transition State Prices We now explain how we use a given spot state price surface to back out one-month time-homogeneous transition state prices. We utilize the property that multiplying spot state prices π (t) 0,i for the maturity indexed by t with the one-month transition probability π i,j accounts for one possible path of getting to state j in time t with corresponding state price π (t ) 0,j, where t indexes the maturity that is month ahead of the maturity indexed by t. Applying the law of total probabilities and

12 summing over all possible paths i leads to the relation: π (t ) 0,j = π (t) 0,i π i,j j, t (8) i I We impose the restriction that π i,j > 0 and use this system of equations to back out transition state prices π i,j in a least squares approach. An identification of transition state prices by using (8) requires the number of used states N has to be smaller than the number of transitions we apply. We estimate monthly transition state prices out of option prices with a maturity of up to one year. Thus, we are limited to eleven nonoverlapping monthly transitions which allows us to only use eleven states to explain the pattern of the spot state price surface and the transition state prices. This demonstrates a problematic feature of implementing the recovery theorem: We need to interpolate the rich implied volatility surface on a rough grid with thirteen points for moneyness (which implies eleven moneyness points for spot state prices) and twelve points for maturity. In addition, we have to make sure that the spot state prices we obtain from this interpolated volatility surface embody tail coverage for high maturity spot state prices as well as fineness for low maturity spot state prices. Ross (205) already mentions this trade-off in an application of his theorem. One general problem here is that recovered physical probabilities could fail to predict future returns just because of this rough discretization. Also, they do not result in accurate approximations of the full spot state price surface. Analogously to Audrino et al. (205), we therefore apply an overlapping approach as our main approach to determine transition state prices. 7 We recall that the maturity of the full spot state price surface was discretized in steps of 3.5 days with a minimum of 6.5 days and a maximum of 365 days. t within 6.5 days and days, we link the corresponding spot state prices π (t) 0,i For each of the maturities at and spot state prices with a maturity at t plus an additional month π (t ) 0,j through one-month transition state prices π i,j. This results in a total number of 94 (overlapping) transitions and thus a maximum number of 7 In the main approach, we use the high-dimensional state space to obtain more detailed results. However, we show in the robustness section 6.A that using coarser state spaces do not alter our main results. 2

13 N =94 allowed states. We then interpolate the implied volatility surfaces equidistantly so that the corresponding spot state prices are defined on 94 points for the moneyness and again make sure that the discretization hits the current state with moneyness=. 8 We back out transition state prices, construct the transition state price matrix Π, and obtain the matrix P= [p i,j ] i,j I of physical transition probabilities by applying the recovery theorem. Figure 4 illustrates the obtained transition state prices and Figure 5 the corresponding physical transition probabilities. We label this version of recovery Ross Basic. The mass of the transition matrix Transition State Prices, 7-Feb Transition state prices State in t State in t + month Figure 4. Transition state prices on the 7-Feb-200 obtained by an implementation without further constraints, only positivity of the transition state prices (basic case). clusters around the main diagonal. Intuitively, this is not surprising, as the probability of moving from state i to the same state i should be higher than the probability of moving from state i to some far away state j. However, the optimization quite often finds large state prices for states that are far away from the current state. This can happen because option prices (in particular, for the near maturities) are hardly affected by such transition state prices in almost irrelevant states. We try to overcome this problem by including further restrictions that are in line with economic theory. 8 Note that we labeled the number of states for the fine discretization with N. For ease of notation, we label the number of states that we use for the interpolation (which is smaller than the number of states for the fine discretization) with N as well. 3

14 Physical Transition Probabilities, 7-Feb-200 Physical Transition Probabilities State in t State in t + month Figure 5. Recovered physical transition probabilities after an application of the recovery theorem (basic case) on the 7-Feb-200. We thus apply an additional approach and force the rows in matrix Π to be unimodal with extreme values on the main diagonal. We further demand all rowsums of Π to lie in the interval [0.9, ]. This ensures reasonable values for the state dependent discount factors and thus the state dependent risk-free rates. Figure 6 shows the transition state prices and Figure 7 shows the recovered physical transition probabilities of this modified approach. By construction, the mass concentrates around the main diagonal without any humps at states that are far away from the initial state. However, for lower (higher) initial states, transition state prices of moving to higher (lower) states are relatively higher than for initial states around the current state. We again think that this is driven by the optimization and the fact that further away states have less importance in explaining option prices. We label this version of recovery Ross Unimodal. 4

15 Transition State Prices (Unimodal), 7-Feb-200 Transition State Prices State in t State in t + month Figure 6. Transition state prices obtained by an implementation with further constraints of unimodality and rowsum bounds (unimodal case) on the 7-Feb-200. Physical Transition Probabilities (Unimodal), 7-Feb-200 Physical Transition Probabilities State in t State in t + month Figure 7. Recovered physical transition probabilities after an application of the recovery theorem (unimodal case) on the 7-Feb

16 C. Recovery without Using Transition State Prices The computation of transition state prices appears to be a key challenge in the application of the recovery theorem. Yet there is one row, the one for the current state, where the transition state prices coincide with the spot state prices for one monthly period, which are much easier to estimate. We use this insight to suggest an alternative method. 9 The trick is using the eigenvalue problem in the recovery theorem as in (6) and multiplying both sides from the left with the transition state price matrix Π: Π Πz = Π δz = δ(πz) = δ 2 z (9) Again, the row of Π 2 associated with the current state contains the spot state prices, but now for two months. Additionally, the discount factor δ now appears in the second power to account for the two months. Iterating, we obtain the following relation: Π t z = δ t z (0) where t determines how often the transition is being applied. A natural property of the transition state price matrix is that Π t again represents a transition state prices matrix, but with a transition period that is t times longer than the transition period of Π. The transition state prices of matrix Π t can again not be inferred directly from option prices. One exception is the row in this matrix that contains transition state prices of moving from the current state of the S&P 500 index to any any other state which simply reflect spot state prices π (t) 0,i with a maturity of t transition periods. We can apply this row property to (0) for different numbers of transition periods t and obtain one equation respectively, which includes known spot state prices with a maturity of t transition periods. We then use this characteristic to identify the pricing kernel implied by z and the utility discount factor δ. In this equation system, the number of unknowns equals to the number of states. A full identification thus requires to use at least as many equations for different maturities t as 9 See the independent derivation in Jensen et al. (206). 6

17 there are number of states N, which results in the following equation system: π () 0, N low π () 0, N low + π () 0,N high π () 0,N high π (2) 0, N low π (2) 0, N low + π (2) 0,N high π (2) 0,N high (T ) (T ) (T ) (T ) π 0, N low π 0, N low + π 0,N high π 0,N high π (T ) 0, N low π (T ) 0, N low + π (T ) 0,N high π (T ) 0,N high z Nlow z 0. z z 0 z z 0. z Nhigh z 0 = δ δ 2. δ T δ T () Appendix C provides more details on how to derive this system of equations. We then solve for δ and z i z 0 by minimizing the sum of squared errors of these N equations under the constraint that the resulting pricing kernel and the utility discount factor δ have to be non-negative. To directly recover one-month pricing kernels and one-month physical probabilities p 0,j, we have to restrict the dimension for moneyness to twelve, as we only can account for a maximum number of twelve transitions and thus twelve maturities. However, we again face the same problem of a rough moneyness grid. 0 We thus implement an alternative approach that allows for a higher state space dimension. To do so, we make use of the property that the structure of the pricing kernel in the setting of Ross (205) remains the same for different maturities and only varies by a factor: m (t) 0,j = δ t m 0,j (2) where m (t) 0,j is the spot pricing kernel with a maturity of t transition periods and m 0,j is the spot pricing kernel with a maturity of one transition period. For our application, we choose a transition period of 3.5 days. This allows us to use a fine moneyness grid with 02 points, as the number of maturities within our maturity window of 6.5 to 365 days increases to N=02. We use (2) to 0 Results and implementation details for this low dimensional approach are given in the robustness section. Appendix C explains in more detail why this property holds. 7

18 find the -month pricing kernel from the 3.5 day pricing kernel. Then we can divide the one-month pricing kernel by the one-month state prices to get the one-month physical spot probabilities. We label this approach Ross Stable. D. Testing the Recovered Probabilities. We test the hypothesis that: H0: Future realized monthly S&P 500 returns are drawn from the recovered distribution. For each day t in our sample, we obtain one realization R t of the monthly S&P 500 future return which is associated with the true pdf p t. Additionally, we recover the physical spot pdf ˆp t and the corresponding cdf ˆP t for that day. Under the assumption ˆp t = p t, the following transformation of R t into u t is i.i.d uniformly distributed, u t i.i.d. U(0, ): 2 u t = ˆP t (R t ) = Rt ˆp t (x)dx (3) We use four different tests. The Berkowitz (200) test jointly tests uniformity and the i.i.d. property of u t. For this test, the series u t is transformed by applying the inverse standard normal cdf Φ to u t : z t = Φ (u t ) = Φ ( Rt ) ˆp t (x)dx (4) Under the hypothesis ˆp t = p t, the z t i.i.d. N(0, ). In the Berkowitz setting, independence and standard normality of z t are tested by estimating the following AR() model: z t µ = ρ(z t µ) + ɛ t (5) 2 See Bliss and Panigirtzoglou (2004) and Cuesdeanu and Jackwerth (206). 8

19 where the null requires µ = 0, Var(ɛ t )= and ρ = 0. Berkowitz suggests a likelihood ratio test with LR 3 = 2(LL(0,, 0) LL(ˆµ, ˆσ, ˆρ)) (6) where LL characterizes the log likelihood of equation (5). We also apply a test recently introduced by Knüppel (205). This test first scales the series u t to y t = 2 (u t 0.5). In order to test u t for standard uniformity, the series y t is tested for scaled uniformity with mean=0 and variance=. The test then compares the first L moments of the series z t to the respective moment of its theoretical counterpart in the following GMM-type procedure with test statistic α L : α L = T D L Ω L D L (7) where D L is a vector that consists of the differences between the sample moments T T t= yl t and the theoretical moments m l for l =,..., L. Ω L is a consistent covariance matrix estimator of all L respective moment differences between y t and its theoretical counterpart. We follow Knüppel (205) and set all elements of Ω L which represent covariances between odd and even moment differences to zero and apply the test by considering the first four moments (L=4). We account for serial correlation of u t by estimating a Newey-West covariance matrix with automated lag length as proposed by Andrews (99). The Berkowitz test has been used in Bliss and Panigirtzoglou (2004). In that study, the authors argue that it is superior to other possible tests such as the Kolmogorov-Smirnoff (KS) test or the Cramer-van-Mises (CvM) test. While being more powerful than KS and CvM, the Berkowitz test bases its conclusion about standard normality on just the first and the second moment of a series, while ignoring higher moments. The test by Knüppel (205) has the advantage of testing for higher moments, too, can also deal with autocorrelated data, and still has enough power to test smaller samples. We perform a simulation study with a sample size of N = 223 to analyze the power of the Berkowitz, the Knüppel, the Cramer-van-Mises, and the Kolmogorov-Smirnoff test. We find that 9

20 the Berkowitz test as well as the test by Knüppel reject the Null hypothesis of standard normality much more likely than the other tests, if we test normally distributed random numbers, but impose a higher or lower variance as in the standard normal case. For our small sample, we therefore rely on the results that we obtain from the Berkowitz test or from the test by Knüppel. We test several versions of recovery for their ability to predict future returns. As a benchmark, we use the distribution of past five years monthly S&P 500 returns, labeled Historical Return Distribution, and a method that assumes a representative investor having a power utility with risk aversion coefficient of 4 for the prediction, labeled Power Utility with γ = 4. 3 Our sample consists of all dates at which options expire in 30 days, which is on the third Friday of the following month. Starting with the first month, we find the percentile u t of the future, realized S&P 500 return under the cumulative recovered physical return distribution function. 4 We then shift to the next month and repeat this procedure. This set of points is then tested for uniformity. We apply that testing approach to all recovery versions and to our benchmark models. IV. Data We use OptionMetrics to obtain end-of-day option data on the S&P 500 index. We also obtain end-of-day S&P 500 index levels from Datastream. We compute S&P 500 returns from January 99 to August 204. Our sample period, as well as our option data, ranges from January 996 to August 204. Consistent with the literature, we only use out-of-the-money put- and call options with a positive trading volume and eliminate all options that violate no arbitrage constraints. We fit the implied dividend yield according to put-call parity where the risk free rate is given by the interpolated zero-curve from OptionMetrics. We consider a monthly horizon and end up with N =223 recovered distributions and realized returns. 3 The results in Bliss and Panigirtzoglou (2004) suggest that a risk aversion factor of 4 in a power utility framework is a reasonable choice. 4 In our implementation, the values of the recovered cdfs ˆP t are given on a discrete grid, whereas the future realized return does typically not lie on the corresponding grid points. We therefore interpolate the recovered cdf linearly to obtain the transformed points u t. 20

21 V. Empirical Results We now present the our empirical findings. We show that the Ross recovery theorem is not suitable to make accurate predictions of future returns and does also not imply plausible economic properties. Unless not stated otherwise, all results are discussed for approaches based on a highdimensional (N=94 if transition state prices are used and N = 02 for Ross Stable) state space. A. In-Sample Fit and Predictive Power of Ross Recovery We use the p-value of the Berkowitz test as the measure for prediction accuracy. In addition to the prediction accuracy, we are able to analyze the in-sample fit of transition state prices. Therefore, we investigate the corresponding row in the transition state price matrix at which the initial state equals the current state. Transition state prices π i,j in this row i should ideally be equal the onemonth spot state prices π 0,j. Paradigmatic for Ross Basic, Figure 8 shows that this is not the case as the spot state prices (dotted) differ from the corresponding transition state prices (gray line). 0.4 State prices (dotted) and transition state prices (grey), 7-Feb Density Value Moneyness Figure 8. Discrepancy between the state prices (dotted) and the transition state prices with the current state as the initially state (gray line) for Ross Basic. Ideally these state price should be the same. 2

22 Furthermore, Figure 9 illustrates the different pricing kernels which we obtain if we either divide spot state prices (dotted line) or transition state prices with initial current state (gray line) by recovered physical spot probabilities. Pricing kernel using state prices and transition state prices, 7-Feb Pricing Kernel Value Moneyness Figure 9. Pricing kernels as the ratio of given spot state prices and recovered physical probabilities (dotted) and as the ratio of transition state prices with initial current state and recovered physical probabilities (gray line) for the basic recovery case. In that sense, we obtain implied distorted state prices ˆπ which comprise an approximation error for each recovery method. For Ross Stable and Power Utility with γ = 4 we do not need transition state prices and thus use interpolated spot state prices with state space dimension N = 02 for ˆπ. Instead of comparing the one-month ˆπ to the true one-month spot state prices π to measure the in-sample fit, we compare the implied volatilities ˆσ associated with ˆπ to observed implied volatilities associated with π. Appendix D provides details for the computation of ˆσ. We define the absolute implied volatility error ɛ l,t as the absolute distance of the approximation ˆσ(M l, t) to its corresponding bid- and ask bounds given by σ bid (M l, t) and σ ask (M l, t) for observation l with moneyness M l and date t: ɛ l,t = max{σ bid (M l, t) ˆσ(M l, t), 0} + max{ˆσ(m l, t) σ ask (M l, t), 0} (8) 22

23 As long as ˆσ(M l, t) moves within bid- and ask bounds, it is ɛ l,t = 0 and there is no approximation error. We compute the mean of the approximation errors ɛ l,t over all sample dates and all observations l and obtain the mean absolute implied volatility (MAIV ) error as a measure for the in-sample performance of each recovery method. Furthermore, we use the p-value of our statistical tests as a measure for forecasting performance. We test additional implementation methods of the recovery theorem which embody more structure than Ross Basic: In Ross Current State, we calculate transition state prices as in Ross Basic but set transition state prices with initial current state equal to the one-month spot state prices. As we pre-set transition state prices with current initial state for this additional recovery method to spot state prices, we compute their corresponding the M AIV errors based on interpolated spot state prices with a state space dimension of N = 94. For Historical Return Distribution we do not apply the M AIV as a second performance measure. Table I shows the p-values and M AIV errors for all introduced methods. Table I: Mean absolute implied volatility errors and p-values of the Berkowitz test and the test by Knüppel for different recovery methods. Recovery Method MAIV Berkowitz: Knüppel: Ross Basic (I): π i,j > 0 Ross Current State (II): π i,j > 0, use spot state prices in Π Ross Unimodal (III): π i,j > 0 and unimodal, rowsums [0.9, ] Ross Stable (IV): Do not use transition state prices error p-value p-value Power Utility with γ = 4 (V) Historical Return Distribution (VI)

24 All tests of recovery methods reject our hypothesis at the 5% level with the Berkowitz test as well as the test by Knüppel. Historical Return Distribution as well as Power Utility with γ = 4 demonstrate a much better forecasting performance. As expected, the MAIV error is low for methods (II) and (IV) as it is in that cases based on interpolated spot state prices. For method (I) we find this error to be significantly higher. The more structure we impose on transition state prices, the worse we expect the in-sample fit to spot state prices. This explains the even higher M AIV error for method (III). B. Economic Insights In theory, the pricing kernel is expected to be positive and monotonically decreasing to reflect the behavior of a risk averse representative investor. Jackwerth (2004), Ait-Sahalia and Lo (2000), and Rosenberg and Engle (2002), however, find that the pricing kernel is locally increasing, a behavior that is referred to as the pricing kernel puzzle. In these papers, physical probabilities are backed out from past S&P 500 index returns, while Ross recovery implies pricing kernels based on forward-looking information. However, Cuesdeanu and Jackwerth (206) confirm the existence of the prizing kernel puzzle in forward-looking data. Figure 4 shows the pricing kernels of different recovery methods for 7-Feb-200. All methods that are associated with Ross recovery (Panel A-D) deliver non-smooth pricing kernels Among all recovery methods, Ross Current State is closest to a monotonically decreasing pricing kernel, while pricing kernels for other recovery methods move rather randomly. Power Utility with γ = 4 implies a monotonically decreasing pricing kernel (Panel E) and is therefore not in line with the pricing kernel puzzle. However, its shape is still more reasonable than the ones we find for almost all of the recovered pricing kernels. 24

25 Panel A: Method (I) Panel B: Method (II) 2 2 Pricing Kernel.5 Pricing Kernel Moneyness Moneyness Panel C: Method (III) Panel D: Method (IV) Pricing Kernel.5 Pricing Kernel Moneyness Moneyness 2.5 Panel E: Method (V) 2 Pricing Kernel Moneyness Figure 0. Pricing kernels for different recovery methods on 7-Feb-200. Panel A shows the pricing kernel for Ross Basic, Panel B for Ross Current State, Panel C for Ross Unimodal, Panel D for Ross Stable, and Panel E for Power Utility with γ = 4. 25

26 We now document further economic implications of the recovery theorem. We first illustrate why state dependent risk-free rates embody information about the recovered pricing kernel: Each rowsum in the transition state price matrix Π represents a state dependent discount factor. The negative relation of the rowsums of Π and the pricing kernel m denotes that a decreasing pricing kernel, which would be in line with economic theory, would require increasing rowsums for Π. This implies a discount factor (risk free rate) that is positively (negatively) related to the state variable, here the S&P 500 index. We furthermore could determine the pricing kernel by just using information on (state dependent) risk-free rates. Martin and Ross (203) demonstrate the impact of interest rates on Ross recovery by showing that the recovered pricing kernel can be inferred from the time series behavior of the long bond and information on the long end of the yield curve. We further illustrate the significance of interest rates for the recovery theorem from another angle. If we assume that risk-free rates are state-independent, all rowsums of Π would be equal. In Ross setting, this implies a constant recovered pricing kernel and thus risk-neutrality. We therefore conclude, that the recovery theorem is incompatible with a simple Black-Scholes economy in which risk aversion is possible despite deterministic risk free rates. There is an even more universal problem if general stochastic models are used to explain the dynamics of the state variable. The recovery theorem assumes that option prices contain information on the future return of the underlying. For any given risk-neutral process, which is fully identified by prices of options on the state variable, we can, however, always assume several stochastic processes that describe the behavior of the state variable itself. Thus, the recovery theorem cannot be verified by using synthetic data without any further assumptions. VI. Robustness A. Dimension of the State Space In a simulation study, Tran and Xia (204) illustrate that the recovered market beliefs are highly dependent on the state space dimension. We now present the implementation and results for the 26

27 low dimensional approaches that limit the number of states to N = (N = 2 for Ross Stable). 5 As we have to deal with the trade-off of tail-coverage for high maturity states and fineness for low-maturity states we implement two different discretization versions for these low-dimensional approaches. In the first version, we interpolate the volatility surface on thirteen equidistant points for moneyness and twelve monthly points for maturity. We make sure that the whole state space range for moneyness is covered and that the discretization hits the current state at moneyness=. To allow for a defined state at moneyness= is necessary, as we recover physical spot probabilities out of physical transition probabilities with current initial state. This first version results in roughly approximated spot state prices for the one-month maturity, but comparably better high-maturity spot state price approximations. In the second version, we still use an equidistant monthly discretization for the maturity, but apply a non-equidistant discretization for the moneyness with more interpolation points around the at-the-money region. We describe how we set these interpolation points in Appendix B. This version provides a more accurate approximation of the one-month spot state prices, but at the cost of a worse approximation of higher maturity spot state prices. We apply the same introduced testing procedure to these low-dimensional approaches. We can reject our hypothesis for all recovery methods at the % level with p-values close to zero and for both equidistant and non-equidistant low-dimensional discretization versions. As the one-month spot state price approximation is more inaccurate with a small state space dimension, the M AIV errors are also significantly higher than for the high-dimensional approach. We further want to check if a variation of the state space dimension for our high-dimensional approaches changes our test results. So far, we used 94 states for an application of methods (I) - (III). We repeat our study, but reduce the dimensions of the state space by about 0% to 84 states 5 For models that use transition state prices, we observe, beginning with month one, non-overlapping monthly transitions for one year of data, which limits the state space to N =. For Ross Stable, we can make use of a maximum of N = 2 states instead, as we use spot state prices with 2 different maturities directly. 27

28 and by about 20% to 74 states, respectively. 6 Table II provides the results for the original state space, after a 0% reduction of the state space dimension, and after a 20% reduction of the state space dimension. We observe that changes in the state dimension do empirically not lead to significant changes for the M AIV pricing error as well as for the Berkowitz test results. The historical return distribution is not affected by the discretization scheme and delivers the same p-values independent of the dimension of the state space. Table II: Mean absolute implied volatility errors and p-values of the Berkowitz test for different recovery methods in the full sample period 0/996-08/204 if the dimension of the state space is not reduced (0%), if it is reduced by about 0%, and if it is reduced by about 20%. Recovery Method Reduction (Number) MAIV Berkowitz: Ross Basic (I): π i,j > 0 Ross Current State (II): π i,j > 0, use spot state prices in Π Ross Unimodal (III): π i,j > 0 and unimodal, rowsum [0.9, ] Ross Stable (IV): Do not use transition state prices Power Utility with γ = 4 (V) of States error p-value 0% (N=94) 0% (N=84) 20% (N=74) 0% (N=94) 0% (N=84) 20% (N=74) 0% (N=94) 0% (N=84) 20% (N=74) 0% (N=02) 0% (N=92) 20% (N=82) 0% (N=02) 0% (N=92) 20% (N=82) So far, we used 02 states for an application of (IV) and model (V). Here, we reduce the state space dimension by about 0% to 92 states and by about 20% to 82 states. 28

29 B. Sub-Samples Our full sample starts in January 996 and ends in August 204 with a sample size of 223 using monthly data. Table III shows the results for the full sample, the sample when excluding the period from 999 until 200 related to the Dot-com bubble, and the sample when excluding the period from 2008 until 2009 related to the financial crisis, respectively. Table III: Mean absolute implied volatility errors and p-values of the Berkowitz test for different recovery methods, if we use data for the full from 0/996 until 08/204, if we exclude the period related to the Dot-com bubble, and if we exclude the period related to the financial crisis, respectively. Recovery Method Sample MAIV Berkowitz: error p-value Ross Basic (I): π i,j > 0 Ross Current State (II): π i,j > 0, use spot state prices in Π Ross Unimodal (III): π i,j > 0 and unimodal, rowsum [0.9, ] Ross Stable (IV): Do not use transition state prices Power Utility with γ = 4 (V) Historical Return Distribution (VI) Full Sample No Dotcom bubble No Financial Crisis Full Sample No Dotcom bubble No Financial Crisis Full Sample No Dotcom bubble No Financial Crisis Full Sample No Dotcom bubble No Financial Crisis Full Sample No Dotcom bubble No Financial Crisis Full Sample No Dotcom bubble No Financial Crisis The statistical tests that we apply are more reliable the larger our sample size is. A large decrease in sample size might therefore itself lead to a change in the test results. We observe a slightly higher p-value for method (III) in the absence of the dot-com bubble period and cannot 29